If the hijacked planes had been shot down, that would have still been a pretty disturbing and traumatizing event - at least a couple of hundred flight crew and passengers dead, and possibly some deaths or injuries on the ground depending on where they're shot down and where the debris lands. Plus, there would be the psychological factor of a situation dangerous enough that the military ends up killing American citizens. I doubt it would have been as extensive, but I could see the U.S. still deciding to help the Northern Alliance and other Afghan opposition forces overthrow the Taliban, for example, or something like the PATRIOT Act passing (probably with less uniform support, though).
If the hijackers had just been caught by airport security or arrested before they got to the airports, then probably not, because it would have been seen as a sign that the safeguards were basically working, rather than that there was a need to ramp up anti-terrorism efforts.